498 CHAPTER 7 Analytic Trigonometry Solving a Trigonometric Equation Using Identities Solve the equation: θ θ θ π + = ≤ < cos sin 2, 0 2 2 Solution EXAMPLE 9 This equation involves two trigonometric functions: sine and cosine. By using a Pythagorean Identity, we can express the equation in terms of just sine functions. θ θ θ θ θ θ ( ) + = − + = − + = cos sin 2 1 sin sin 2 sin sin 1 0 2 2 2 cos 1 sin 2 2 θ θ = − This is a quadratic equation in θ sin . The discriminant is b ac 4 1 4 3 0. 2 − = − =− < Therefore, the equation has no real solution. The solution set is the empty set, ∅. Check: Graph = + Y x x cos sin 1 2 and = Y 2 2 to see that the two graphs never intersect, so the equation = Y Y 1 2 has no real solution. See Figure 30. Figure 30 0 22 2p 3 Y1 5 cos 2x 1 sin x Y2 5 2 Solving a Trigonometric Equation Using a Graphing Utility Solve: + = x x 5sin 3 Express the solution(s) rounded to two decimal places. Solution EXAMPLE 10 This trigonometric equation cannot be solved by previous methods.A graphing utility, though, can be used. Each solution of the equation is the x -coordinate of a point of intersection of the graphs of = + Y x x 5sin 1 and = Y 3. 2 See Figure 31. There are three points of intersection; the x -coordinates are the solutions of the equation. Use INTERSECT to find = = = x x x 0.52 3.18 5.71 The solution set is { } 0.52, 3.18, 5.71 . Figure 31 Y1 5 5 sin x 1 x 4p 2p Y2 5 3 28 14 5 Solve Trigonometric Equations Using a Graphing Utility The techniques introduced in this section apply only to certain types of trigonometric equations. Solutions for other types are usually studied in calculus, using numerical methods. Now Work PROBLEM 85 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 7.3 Assess Your Understanding 1. Find the exact value of π π − sec 15 tan 15 . 2 2 (pp. 417–419) 2. π( ) = sin 4 ; π ( ) = cos 8 3 (pp. 398–405) 3. Find the real solutions of − − = x x 4 5 0. 2 (pp. A47–A53) 4. Find the real solutions of − − = x x 1 0. 2 (pp. A47–A53) 5. Find the real solutions of ( ) − − − − = x x (2 1) 3 2 1 4 0. 2 (pp. A53–A54) 6. Use a graphing utility to solve − = − x x x 5 2 . 3 2 Round answers to two decimal places. (pp. 28–31) Concepts and Vocabulary 7. True or False Most trigonometric equations have unique solutions. 8. True or False Two solutions of the equation θ = sin 1 2 are π π 6 and 5 6 . 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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